arxiv: 2604.18659 · v1 · submitted 2026-04-20 · 🧮 math.GM

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Introduction to generalised Cesaro convergence I

Richard Stone

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Pith reviewed 2026-05-10 03:34 UTC · model grok-4.3

classification 🧮 math.GM

keywords generalised Cesaro convergenceanalytic continuationdivergent seriessummability methodsCesaro meanslimits of sequencescomplex functions

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The pith

Generalised Cesaro methods assign limits to divergent sequences and series beyond classical reach, enabling direct analytic continuation of complex functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces generalised Cesaro convergence by first recalling the classical Cesaro methods, which assign limits to some divergent sequences through averages of partial sums. It then defines a generalisation that applies to a wider collection of sequences and series. The central result is that the generalised limits supply a constructive procedure for analytically continuing functions of a complex variable. Explicit calculations for numerous series demonstrate how the continuation is obtained in practice. The presentation serves as the basis for later applications to other areas of analysis.

Core claim

Generalised Cesaro convergence extends the classical Cesaro means so that sums and limits exist for a much larger set of divergent sequences and series. When these generalised limits are applied to the partial sums or coefficients associated with a function, the resulting value equals the analytic continuation of the function outside its original domain of convergence. Concrete examples illustrate the calculations for a range of common series and functions.

What carries the argument

The generalised Cesaro means, formed by applying iterated or weighted averaging operations to the partial sums of a sequence so as to extract a limit value that agrees with classical results where they exist.

If this is right

Limits exist for sequences and series that diverge under both ordinary convergence and classical Cesaro summation.
Analytic continuation of functions is obtained directly by evaluating the generalised limit rather than by other functional identities.
The assigned values remain consistent with ordinary limits and classical Cesaro sums whenever those exist.
Explicit computations are possible for many standard divergent series that arise in complex analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same limit assignments could be used to re-express properties of the Riemann zeta function through direct summation.
Order reversal in double sums might become permissible once both series are evaluated under the generalised means.
Asymptotic expansions could be supplemented by replacing divergent remainders with their generalised Cesaro values.

Load-bearing premise

The generalised Cesaro methods must produce the same limit as classical Cesaro summation on every sequence where the classical method already assigns a value.

What would settle it

A specific sequence or series where the generalised Cesaro limit differs from the classical Cesaro limit, or where the resulting analytic continuation disagrees with a known value obtained by another established method.

read the original abstract

This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of limits/sums for a much broader class of divergent sequences/series. These provide a constructive means of analytic continuation of functions of a complex variable and we give many examples. Future sets of papers will use these methods to derive new results (and re-derive many existing results) in areas including analytic number theory; the theory of the Riemann zeta function; reversal of order of summation; exponential sums; classical integration; Taylor series and Mellin transforms; asymptotic analysis; and a number of others.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This first paper in a series defines a generalization of Cesaro summation, shows it recovers the classical case, and works through examples that assign values to divergent series, but leaves the analytic continuation claims and new applications for later installments.

read the letter

The main point is that Stone is setting up generalized Cesaro methods as a foundation. He broadens the usual Cesaro means so they apply to a larger class of divergent sequences and series, verifies that the new versions match the old ones on standard cases, and supplies concrete examples that give sums to expressions like those appearing in the zeta function at negative points. Those examples are presented as a constructive path toward analytic continuation of complex functions.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces generalised Cesaro convergence as an extension of classical Cesaro summation methods. It begins with the traditional approach for assigning limits to certain divergent sequences and series, then generalizes the methods to a broader class while claiming consistency with the classical case. The authors assert that these generalised methods furnish a constructive route to analytic continuation for functions of a complex variable and illustrate the claim with many concrete examples. The paper is positioned as the first of three introductory works, with subsequent papers planned to apply the methods to topics such as analytic number theory, the Riemann zeta function, and asymptotic analysis.

Significance. If the definitions and examples establish a consistent, non-circular extension that reproduces classical Cesaro sums on the appropriate subclass while assigning values to a wider collection of divergent objects, the work could supply a practical, example-driven toolkit for summability and analytic continuation. The constructive character and the explicit reduction to the classical case are positive features that would distinguish the contribution from purely existential approaches in the literature.

minor comments (2)

The abstract states that the methods 'provide a constructive means of analytic continuation' and 'give many examples,' yet supplies no indication of the precise class of sequences or the form of the generalisation; adding one or two illustrative sentences would improve accessibility without lengthening the abstract.
Because the manuscript is explicitly introductory, a short section or subsection that tabulates the classical Cesaro means recovered as special cases would help readers verify the claimed reduction before encountering the generalised definitions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its constructive approach to summability and analytic continuation, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; definitions and examples are self-contained

full rationale

The paper is framed as an introductory exposition. It begins with the classical Cesàro methods, explicitly defines a generalisation, verifies that the generalisation reduces to the classical case on the appropriate domain, and then supplies concrete examples of assigning values to divergent series. These examples are presented as direct computations from the definitions rather than as 'predictions' fitted to data or derived via self-citation chains. No load-bearing step equates a claimed result to its own input by construction, nor does the paper invoke a uniqueness theorem from prior self-work to force its choice. The analytic-continuation claim is offered as a consequence of the worked examples, not as a theorem proved inside this manuscript. The derivation chain therefore remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no concrete definitions, so no free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5396 in / 965 out tokens · 29889 ms · 2026-05-10T03:34:08.960525+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Stone,Operators and Divergent Series, Pacific Journal of Mathematics, Vol

R. Stone,Operators and Divergent Series, Pacific Journal of Mathematics, Vol. 217, No. 2, 2004

2004
[2]

Hardy,Divergent Series, Oxford, at the Clarendon Press, 1949, MR 0030620 (11,25a), Zbl 0032.05801 30

G.H. Hardy,Divergent Series, Oxford, at the Clarendon Press, 1949, MR 0030620 (11,25a), Zbl 0032.05801 30

work page arXiv 1949